Melt the butter in a 10-inch nonstick skillet over medium heat. Set aside 2 Tbs. of the melted butter in a small bowl. Add the mushroom stems and 1/4 tsp. salt to the skillet and cook, stirring often with a wooden spatula, until tender, about 5 minutes. Let cool to room temperature.

Meanwhile, add 2 Tbs. of the Parmigiano and the panko to the reserved melted butter and mix thoroughly; set aside.

In a medium bowl, stir together the remaining Parmigiano, the mayonnaise, parsley, and 1/4 tsp. black pepper. Add the mushroom stems and mix until well combined.

In a large bowl, toss the mushroom caps with the oil. Fill each mushroom cap with a rounded teaspoon of the filling and top each with about 1 tsp. of the panko mixture. Place on a large rimmed baking sheet with space in between. Bake until tender and the topping is golden-brown, about 25 minutes. Cool briefly before serving.

Make Ahead Tips

The mushrooms can be stuffed up to 2 hours ahead; refrigerate, covered, until ready to bake.